Characterization and Analysis of Multi-Hop Wireless MIMO Network Throughput

Transcription

1 Characterzaton and Analyss of Mult-Hop Wreless MIMO Network Throughput Bechr Hamdaou EECS Dept., Unversty of Mchgan 226 Hayward Ave, Ann Arbor, Mchgan, USA Kang G. Shn EECS Dept., Unversty of Mchgan 226 Hayward Ave, Ann Arbor, Mchgan, USA ABSTRACT Use of multple antennas or MIMO has great potental for enhancng the throughput of mult-hop wreless networks va spatal reuse and/or spatal dvson multplexng. In ths paper, we characterze and analyze the maxmum achevable throughput n mult-hop wreless MIMO networks under three MIMO protocols, spatal reuse only (), spatal multplexng only (), and spatal reuse & multplexng (), each of whch enhances throughput va a dfferent way of explotng the MIMO s potental. We show va extensve smulaton that as the number of antennas ncreases, the maxmum achevable throughput frst rses and then flattens out asymptotcally under, whle t ncreases almost" lnearly under or. We evaluate the effects of several network parameters on ths achevable throughput. We also demonstrate how these results can be used by desgners to determne the optmal parameters of mult-hop wreless MIMO networks. Categores and Subject Descrptors: I [Smulaton and Modelng]: Smulaton Output Analyss General Terms: Performance, Desgn Keywords: Network throughput analyss, MIMO systems, multhop wreless networks, wreless mesh networks. INTRODUCTION Multple antennas, also referred to as MIMO (multple-nputmultple-output), provde wreless networks wth potental for ncreasng network throughput va spatal reuse of the spectrum by allowng multple smultaneous communcaton sessons n the same neghborhood and/or va spatal dvson multplexng by achevng hgh data rates. For ths reason, MIMO systems are expected to be a key component of next-generaton wreless networks. From the physcal layer s standpont, the potental benefts of multple antennas are already well-understood [ 5]. How to realze these benefts at hgher layers has also been studed recently [6 ]. These studes focused on the development of MAC protocols for wreless networks that explot multple antennas to ncrease the overall network throughput va spatal reuse [, ] and/or spatal multplexng [7], or reduce power consumpton va beam-formng The work reported here was supported n part by NSF under Grant No. CNS and Intel Corporaton. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. MobHoc 7, September 9 4, 27, Montréal, Québec, Canada. Copyrght 27 ACM /7/9...$5.. and nterference suppresson [6]. However, how much throughput multple antennas can offer mult-hop wreless networks has been studed much less [2]. Y et al. [2] extended the work n [3] to wreless networks usng drectonal antennas. The focus n [2] s, however, on the swtched mult-beam technque. Albet smple, the swtched mult-beam technque works only n a near lne-of-sght envronment, and may ncrease the capacty only through spatal reuse. In ths paper, we characterze and analyze the maxmum achevable throughput n mult-hop wreless MIMO networks when the adaptve array technque s used. Unlke the swtched multbeam technque, the adaptve array technque can explot multple antennas to ncrease the capacty n both lne-of-sght and multpath envronments [4] va not only spatal reuse but also spatal multplexng. The man contrbutons of ths paper are summarzed as follows.. Modelng of the nterference and rado constrants on multhop wreless MIMO networks under the three MIMO protocols and two nterference avodance models we propose. 2. Characterzaton and analyss of the maxmum achevable throughput n mult-hop wreless MIMO networks. Va extensve smulatons, we show that as the number of antennas ncreases, the maxmum achevable throughput flattens out asymptotcally under and ncreases almost" lnearly under or. 3. Evaluaton of the effects of several network parameters on ths achevable throughput. We also demonstrate practcal use of the obtaned results by llustratng how they can be used by network desgners to determne the optmal parameters of mult-hop wreless MIMO networks. The rest of ths paper s organzed as follows. Secton 2 dscusses the related work, puttng our work n a comparatve perspectve. Secton 3 overvews MIMO and llustrates ts potental benefts. We model the network under study and state our objectves n Secton 4. Secton 5 models the packet-level constrants, whle Secton 6 formulates the mult-commodty flow routng problem. Throughput characterzaton and analyss are provded n Secton 7. Fnally, we conclude the paper n Secton RELATED WORK There have been numerous studes on throughput/capacty characterzaton of wreless networks equpped wth sngle antennas [3, 5 8]. Gupta and Kumar [3] derved the asymptotc capacty of mult-hop wreless networks of statc nodes, each equpped wth a sngle omndrectonal antenna. The work n [5] shows that peruser throughput can ncrease dramatcally when nodes are moble rather than fxed by explotng a form of multuser dversty va

2 packet relayng. Several other studes have also focused on characterzng the capacty n mult-channel wreless networks [6 8]. The work n [3] has been extended n [6] to mult-channel wreless networks where nodes, each equpped wth multple nterfaces, cannot have a dedcated nterface per channel. Ther results show that the capacty of such networks depends on the rato of the number of channels to the number of nterfaces. Alcherry et al. [7] developed a soluton for routng n mult-channel, mult-nterface wreless mesh networks that maxmzes the overall throughput of the network subject to farness and nterference constrants. Along the same lne, the work n [8] provdes necessary condtons for the feasblty of rate vectors n mult-channel wreless networks wth multple nterfaces, and use them to fnd upper bounds on throughput va a fast prmal-dual LP algorthm. We adapt the LP constrant relaxaton technque from [8] to characterze the maxmum achevable throughput n mult-hop wreless networks of nodes equpped wth MIMO lnks. s (t) s (t) 2 s(t) Transmtter Recever 2 u u 2 u u u u, 2,,2 2,2 (a) A MIMO Lnk (b) One-stream sgnal. (c) Two-stream sgnal. v v v v, 2,,2 2,2 v v 2 r(t) r (t) r (t) 2 3. PRELIMINARIES: MIMO LINKS The term MIMO lnk s used to denote any transmtter-recever par such that () the recever s wthn the transmtter s transmsson range, and (2) both the transmtter and recever are equpped wth multple antennas. 3. Bascs of MIMO Let s consder the MIMO lnk shown n Fg. (a), and assume that the transmtter and the recever are each equpped wth 2 antennas. To transmt a sgnal s(t) over the 2-antenna array, the transmtter sends two weghted copes, u s(t) and u 2 s(t), of the sgnal, one on each antenna; the vector u = [u u 2] T s referred to as a transmsson weght vector. At the recever, the two receved sgnals (one on each antenna) are weghted wth a recepton weght vector v = [v v 2 ] T and summed to produce r(t). Ths s llustrated n Fg. (b). Let H denote the matrx of channel coeffcents between the transmtter and the recever. One can then wrte r(t) = (u T Hv)s(t). By choosng approprate weght vectors u and v, one can ensure that the sgnal r(t) acheves a unt gan (u T Hv = ) when receved by the target recever, and a zero gan (u T Hv = ) when receved by a non-target recever. Hence, wth multple antennas, a node can successfully communcate wth ts target recever whle allowng other nearby recevers to successfully receve ther sgnals. Multple antennas can also be exploted to send multple-stream sgnals. As shown n Fg. (c), the transmtter can send two streams, s (t) and s 2(t), each weghted over both antennas usng the transmsson weght vectors u = [u, u,2 ] T and u 2 = [u 2, u 2,2 ] T, respectvely. At the recever, two separate streams, r (t) and r 2 (t), are constructed by weghtng the two receved sgnals (one on each antenna) by two recepton weght vectors v = [v, v,2] T and v 2 = [v 2, v 2,2 ] T. One can wrte r (t) = (u T Hv )s (t) + (u T 2 Hv )s 2 (t) and r 2 (t) = (u T Hv 2 )s (t) + (u T 2 Hv 2 )s 2 (t). Wth an approprate choce of all the weght vectors and under the assumpton that H s a full-ranked matrx [5], one can ensure that u T Hv = and u T 2 Hv = to correctly construct r (t), and u T Hv 2 = and u T 2 Hv 2 = to correctly construct r 2 (t). Hence, multple antennas can be exploted to ncrease the data rates by sendng multple-stream sgnals. 3 Benefts of MIMO To llustrate MIMO benefts, let s consder the example of a multhop MIMO network n Fg. 2, whch conssts of a set N = {, 2, 3, 4} The superscrpt T ndcates the matrx transpose operaton. Fgure : MIMO processng. f 2 f Fgure 2: An llustratve network example. of 4 nodes, and a set L = {(, 3), (2, 4), (, 4)} of MIMO lnks. Suppose each node has 2 antennas (γ m = 2, m N). 3. Spatal Reuse Due to multple antennas, transmtters can null ther sgnals at undesred nearby recevers (.e., prevent ther sgnals from reachng undesred nearby recevers) whle ensurng acceptable sgnal gans at ther desred recevers. Lkewse, recevers can use ther multple antennas to suppress nterferences caused by undesred nearby transmtters whle successfully recevng ther desred sgnals. For the purpose of llustraton, let s assume that, at a gven tme t, nodes and 2 both decded to transmt sgnals to nodes 3 and 4, respectvely. Frst, note that f nodes are equpped wth sngle omndrectonal antennas, then node s transmsson wll nterfere wth node 4 s recepton, and hence, node 4 won t be able to successfully receve the sgnal from node 2. Because node 4 has 2 antennas, ts recepton weght vector v 4 can be so chosen that the nterference caused by node s transmsson may be suppressed whle assurng an acceptable gan of ts ntended sgnal from node 2. These constrants or requrements can be wrtten as (u T 2 H 2,4 )v 4 = and (u T H,4)v 4 = where u 2 = [u 2, u 2,2] T s the transmsson weght vector of node 2 and v 4 = [v 4, v 4,2] T s the recepton weght vector of node 4. Knowng H,4, H 2,4, u, and u 2, node 4 can solve the system of these two equatons to determne v4 whch can then be used to receve an nterference-free sgnal from node 2 concurrently wth node s transmsson sgnal. Multple antennas can thus be exploted to ncrease spatal reuse by allowng multple smultaneous transmssons n the same vcnty. 3 Spatal Dvson Multplexng Suppose node does not transmt at tme t, then node 4 can use both antennas to receve two streams of data concurrently. To desgn ts recepton weght vectors v 4, = [v 4,, v 4,,2 ] T and v 4,2 = [v 4,2, v 4,2,2 ] T, we need to solve two systems of lnear equatons { (u T 2, H 2,4 )v 4, = (u T 2,2H 2,4 )v 4, = and { (u T 2, H 2,4 )v 4,2 = (u T 2,2H 2,4 )v 4,2 = where u 2, = [u 2,, u 2,,2 ] T and u 2,2 = [u 2,2, u 2,2,2 ] T are the

3 two transmsson weght vectors used by node 2 to transmt ts two streams. The soluton can then be used by node 4 to receve two concurrent data streams from node 2. Hence, multple antennas can also be used to ncrease the transmsson rates by explotng the spatal multplexng offered by the antennas. Note that now, node cannot transmt wthout causng nterference at node 4; spatal reuse cannot be ncreased when all antennas are used for spatal multplexng. 3 Interference Avodance Models We now propose two models 2 that can be used by nodes to suppress nterference and/or null undesred sgnals so that the spatal reuse of spectrum may be ncreased. Non-Cooperatve Interference Avodance Model (NM): Ths model requres that () transmtters be responsble for nullng ther sgnals at all nearby nterferng recevers pror to transmttng ther sgnals, and (2) recevers be responsble for suppressng the nterference caused by all nearby transmtters pror to recevng ther desred sgnals. That s, before transmttng ts sgnal, a transmtter must ensure that t has enough antennas to transmt the sgnal wthout causng nterference to any of ts nearby recevers. Lkewse, pror to recevng sgnals, a recever must ensure that t has enough antennas to be able to suppress the nterference caused by all nearby transmtters whle recevng ts desred sgnals wthout nterference. In the example network of Fg. 2, under NM, node 4 must then be able to suppress node s sgnal pror to recevng node 2 s sgnal, and node must be able to null ts sgnal at node 4 pror to transmttng a sgnal to node 3. Cooperatve Interference Avodance Model (CM): Note that t suffces for node 4 to suppress node s sgnal, or for node to null ts sgnal at node 4 to have two successful transmssons. Unlke NM, CM requres that ether the transmtter or the recever (not necessarly both) be responsble for nterference avodance. Referrng to the example of Fg. 2 agan, nodes and 4 must then coordnate to desgn ther vectors such that u T (H,3v 3) = (ensured by node ) u T H,4v 4 = (ensured by ether node or node 4) (u T 2 H 2,4 )v 4 = (ensured by node 4). Clearly, CM provdes hgher spatal reuse of multple antennas than NM. Ths wll be justfed later. 3 Effectve Degrees of Freedom Based on the llustratons gven n Secton 3, one can draw the followng concluson. A node s degrees of freedom (DoFs or number of antennas) can be exploted n one of the followng three ways: () all DoFs are used to send a multple-stream flow of data by explotng the spatal dvson multplexng of the antenna array; (2) all DoFs are used to ncrease the spatal reuse of the spectrum by allowng multple concurrent streams n the same vcnty; (3) some of DoFs are used to send a multple-stream flow whle the others are used to allow for concurrent streams n the same neghborhood. It s mportant to note that the level of explotaton of the spatal reuse and/or multplexng s, however, contngent on physcal lmtatons such as node s power, multpath, and/or channel coeffcents estmaton errors [9]. Let s consder two neghbor nodes m and n each equpped wth an antenna array of sze γ m and γ n, respectvely, and assume that m wants to transmt a χ-stream data sgnal to n. Suppose there are ϕ streams currently beng receved by nodes located wthn m s 2 It s mportant to menton that we only provde key features of the models relevant to ths work. Hence, we omt detals on how and when nodes exchange nformaton such as weght vectors. transmsson range, and ψ streams currently beng transmtted by nodes located wthn n s recepton range. Due to physcal lmtatons, the number (ϕ + χ) of possble concurrent streams n m s vcnty s lkely to be less than the number of ts actual antenna elements γ m [9]. We wll refer to ths number α m = (ϕ + χ) as effectve transmt degrees of freedom of node m. For smlar reasons, the number (ψ + χ) of possble concurrent streams n n s vcnty s also lkely to be less than ts total number of antennas γ n [9]. Ths number β n = (ψ + χ) wll be referred to as effectve receve degrees of freedom of node n. The authors of [2] derved a statstcal method that allows each node m to determne both α m and β m gven the network s physcal constrants. In ths paper, we assume that these two numbers are known for each node. 4. PROBLEM STATEMENT In ths secton, we model mult-hop wreless MIMO networks, state our objectves, and outlne how to meet them. 4. Network Model: Assumptons & Notaton A mult-hop wreless MIMO network s modeled as a drected graph G = (N, L) wth a fnte nonempty set N of nodes and a fnte set L of MIMO lnks. L s the set of all ordered pars (m, n) of dstnct nodes n N such that n s wthn m s transmsson range. If lnk = (m, n) L, then node m and node n are referred to as the transmtter t() and the recever r() of lnk. A data lnk s sad to be actve f t() s currently transmttng to r(); otherwse, t s sad to be nactve. For every m N, let L + m = { L : t() = m} denote the set of all lnks whose transmtter s m, L m = { L : r() = m} denote the set of all lnks whose recever s m, and L m = L + m L m. We assume that each node m s equpped wth an antenna array of γ m elements t uses to transmt and receve sgnals. Let α m and β m denote node m s effectve transmt and receve degrees of freedom. For every L, let c denote the maxmum number of bts that lnk can support n one second. Whle c depends on (.e., could vary from lnk to lnk), t s assumed to be tme-nvarant. Let C be the set of all ordered dstnct pars (, j) L L such that and j do not share a node between them and the transmsson on lnk nterferes wth the recepton on lnk j. Note that (, j) C does not necessarly mply that (j, ) C. Gven a lnk L, let C + = {j L : (, j) C} denote the set of all lnks whose recevers nterfere wth the transmsson on, and C = {j L : (j, ) C} denote the set of all lnks whose transmtters nterfere wth the recepton on. 4 Objectves, Approaches, & Contrbutons We want to characterze and analyze the maxmum achevable throughput n mult-hop wreless MIMO networks. We propose and analyze three dfferent MIMO protocols spatal reuse only protocol (), spatal multplexng only protocol (), and spatal reuse & multplexng protocol () all of whch ncrease network throughput, but each wth a dfferent way of explotng the multple antenna benefts. Spatal Reuse Only MIMO Protocol (): uses all effectve degrees of freedom to ncrease network throughput va spatal reuse of the spectrum only. In, the throughput s then ncreased by allowng multple smultaneous communcaton sessons n the same neghborhood. Spatal Multplexng Only MIMO Protocol (): under whch all effectve DoFs are used to ncrease throughput va spatal multplexng only. Nodes n can use ther multple antennas to communcate multple stream sgnals among them. They cannot, however, use any of ther effectve DoFs to ncrease spatal reuse.

4 Spatal Reuse & Multplexng MIMO Protocol (): s a combnaton of and n that the effectve degrees of freedom can be used to ncrease network throughput va spatal reuse and/or spatal multplexng, whchever provdes hgher throughput. We consder TDMA n whch tme s dvded nto tme slots of an equal length, denoted by T = {, 2,...}. Characterzng the achevable throughput under TDMA wll then serve as a characterzaton of the throughput achevable under other multple access methods, such as CDMA and CSMA/CA. For each MIMO protocol, we formulate the mult-hop routng problem as a standard mult-commodty flow nstance that conssts of a set Q of commodtes where each q Q s characterzed wth a source-destnaton par s(q), d(q) N and a non-negatve multhop flow of rate f q. A mult-hop flow soluton maxmzng the sum q Q f q of all flows rates subject to the network constrants that we wll descrbe and model n next sectons wll be used to represent the achevable throughput under mult-commodty flow f = (f q) q Q. By solvng many nstances, we can provde a statstcal characterzaton and analyss of the maxmum achevable throughput n mult-hop wreless MIMO networks. Our contrbuton s twofold. Frst, we characterze and analyze the maxmum achevable throughput n mult-hop wreless networks equpped wth MIMO systems. We study the effects of several network parameters on ths throughput. Second, we show how the thus-obtaned results can be used for desgnng wreless MIMO networks such as MIMO mesh networks. These results enable network desgners to determne the optmal parameters of wreless MIMO networks. 5. PACKET-LEVEL CONSTRAINTS We now model the packet-level constrants on mult-hop MIMO networks, descrbed n Secton 4. (, t) L T, let the bnary varable y t be f lnk s actve durng tme slot t, and otherwse. 5. Spatal Reuse Only MIMO Protocol () 5.. Rado Constrants Due to rado lmtatons, we assume that a node can ether transmt or receve, but not both, at a tme slot. Also, snce explots all degrees of freedom (DoFs) to ncrease spatal reuse, a node can use at most one DoF to transmt or receve one stream whle the other DoFs can be used to allow for multple concurrent streams n same vcntes. Hence, one can wrte L m y t, m N, t T. () 5. Interference Constrants Next we descrbe the nterference constrants under both the noncooperatve nterference avodance model (NM) and the cooperatve nterference avodance model (CM), as defned n Secton 3. Interference Constrants under NM: Recall that under NM, recevers must be responsble for suppressng sgnals from nterferng transmtters. Hence, any recever must have enough effectve receve degrees of freedom that enable t to combat nearby transmtters nterference pror to recevng a sgnal at any tme slot. That s, L and t T, (ω β r() + )y t + j C y t j ω (2) where ω s an nteger larger than the maxmum number of actve lnks at any gven tme slot. Let ω = L. If y t = (.e., s actve at tme slot t), then the above constrants ensure that the total number of actve lnks, nterferng wth the recepton on lnk, does not exceed what node r() s effectve receve degrees of freedom can handle; otherwse (f y t = ), the constrants are relaxed snce s not actve, and hence, no nterference needs to be suppressed. Lkewse, transmtters under NM must also be responsble for nullng ther sgnals at all nearby recevers. That s, pror to transmsson at any tme slot, a transmtter must have enough effectve transmt degrees of freedom so that t can prevent ts sgnal from causng nterference to any nearby recevers. Hence, we can wrte, for all L and all t T, (ω α t() + )y t + j C + y t j ω. (3) Agan, the above constrants ensure that the maxmum number of actve lnks nterferng wth the transmsson on lnk does not exceed what node t() can null,.e., no more than α t() can be concurrently actve at tme slot t when s actve. If, however, t() s not transmttng (.e., y t = ), then the constrants are relaxed as expressed by the nequalty va ω. Interference Constrants under CM: Under CM, for every par (, j) C, one of the followng two condtons must hold: the transmtter of must null ts sgnal at the recever of j; or the recever of j must suppress the nterference from the transmsson on lnk. Note that one (and only one) of the above two condtons needs to hold for a successful transmsson on whle stll recevng an nterference-free sgnal on j. To express ths set of constrants, we need to ntroduce two new bnary varables. For every t T and for every (, j) C, we defne bnary varables λ t j = f and j are both actve at t, and t() nulls ts sgnal at r(j) otherwse and bnary varables f and j are both actve at t, and r(j) µ t j = suppresses the nterference from t() otherwse. The nterference constrants to under CM can then be expressed as follows. For all (, j) C and all t T, + l C + λ t l α t() + l C µ t lj β r(j) (4) j y t + yj t λ t j + µ t j +. 5 Spatal Multplexng Only MIMO Protocol () 5. Rado Constrants Recall that explots all DoFs to ncrease throughput by allowng transmtter-recever pars to communcate multple stream sgnals over ther lnks,.e., each transmtter-recever par, (t(), r()), can communcate more than one stream over lnk. Let z t represent the number of streams that are actve on lnk at tme slot t. Because the maxmum number of streams communcated on lnk must not exceed the effectve transmt degrees of freedom of t() nor the effectve receve degrees of freedom of r(), z t α t() y t and z t β r() y t (5) must hold L and t T. Lke n, n, a node can ether transmt or receve at any gven tme slot, and can at most be actve on one lnk. Hence, the constrants n Eq. () must also hold under ;.e., L m y t, m N, t T. (6)

5 5 Interference Constrants Recall that all DoFs n are used for spatal multplexng,.e., none of them are exploted to ncrease spatal reuse. Therefore, NM and CM are equvalent under, and so are the nterference constrants. These constrants can be wrtten as y t + y t j, (, j) C, t T. (7) 5 Spatal Reuse & Multplexng MIMO Protocol () We now descrbe and model the packet-level constrants under. Note that the rado constrant under are equvalent to those under as descrbed n Secton 5.. The nterference constrants, however, are dfferent from those under or. Interference Constrants under NM: Under NM, recevers are responsble for suppressng sgnals from nterferng transmtters,.e., for all L and all t T, (Ω β r() )y t + j C z L j t Ω (8) r() and transmtters are responsble for nullng ther sgnals at all nearby recevers,.e., for all L and all t T, (Ω α t() )y t + j C + z L+ j t Ω (9) t() where Ω s an nteger greater than the number of possble concurrent streams. Let Ω = L max m N γ m. Interference Constrants under CM: For every (, j) C and for every t T, we ntroduce two nteger varables, θj t and ϑ t j. θj t represents the number of DoFs assgned by t() to null ts sgnal at r(j), provded both and j are actve,.e., r(j) can have up to θj t nterference-free streams. ϑ t j represents the number of DoFs assgned by r(j) to suppress nterference comng from t(), provded both and j are actve,.e., ϑ t j streams can be sent by t() wthout causng nterference at r(j). The constrants under CM can then be wrtten as follows. (, j) C and all t T, l L + zl t + l C + θl t α t(), t() l L zl t + l C ϑ t lj β r(j), r(j) j () z t ϑ t j + α t() ( y), t zj t θj t + β r(j) ( yj). t 5 Observatons There are two ponts worth mentonng regardng the above desgn constrants. Frst, they all constran the feasblty of data transmssons on a packet-by-packet bass. That s, at every tme slot, packet-level condtons must all be met n order for packet transmssons to be successful durng that tme slot; these constrants can then be seen as condtons under whch the nstantaneous lnk rates are feasble. Second, they all are necessary condtons, but not suffcent for the feasblty of packet transmssons. That s, f, at a gven tme slot t, some or all of these constrants are not met, then some or all of the packets transmtted at tme t wll be unsuccessful, whereas meetng all of these constrants does not guarantee successful transmssons of all packets. 6. MULTI-COMMODITY FLOW 6. LP Relaxatons: Flow-Level Desgn There are two subtle ssues wth the packet-level constrants descrbed n Secton 5. Frst, they are expressed n nteger varables. Hence, the mult-commodty flow formulaton descrbed n Secton 4 cannot be solved by the standard lnear programmng. Second, they are nstantaneous,.e., at every tme slot, there s a set of constrants that must be met. Ths wll ncrease the sze of the optmzaton problem n terms of both the number of constrants and the number of varables. We want to provde LP relaxatons of these constrants to address the above two ssues. As t wll become clear shortly, the relaxed constrants can be seen as necessary condtons on the feasblty of average lnk rates. Note that, by defnton, LP relaxatons result n wdenng the feasblty space; that s, the solutons obtaned under the average-rate (relaxed) constrants may be nfeasble under the nstantaneous-rate constrants. However, snce we am to characterze the maxmum achevable throughput, these relaxatons wll only make the maxmum less tght. Clearly, there s a tradeoff between the qualty of solutons and the sze/complexty of problems. To keep the problem smple whle drawng useful conclusons, we choose to work wth the relaxed constrants nstead of the packetlevel ones. Next we provde LP relaxatons to the packet-level constrants descrbed n the prevous secton. Let s consder a set of tme slots S T of cardnalty τ, and for all L, defne y to be τ t S yt. For every (, j) C, let λ j = τ t S λt j and µ j = τ t S µt j. Note that y represents the fracton of tme n S durng whch lnk s actve; λ j represents the fracton of tme n S durng whch lnks and j are both actve and t() s nullng ts sgnal at r(j); and µ j represents the fracton of tme n S durng whch lnks and j are both actve and r(j) s suppressng the nterference caused by t() s sgnal. For every L, we also defne the contnuous varables z as τ t S zt, and for all (, j) C, let θ j = τ t S θt j and ϑ j = τ t S ϑt j. Suppose that, j L are both actve durng S. Here, z represents the average number of streams that are actve on lnk durng S; θ j represents the average number of effectve transmt degrees of freedom that t() allocates to null ts sgnal at r(j); and ϑ j represents the average number of effectve receve degrees of freedom that r(j) allocates to suppress the nterference comng from t(). Recall that all these contnuous varables are averages over the length of the tme slot set S. Hence, the longer S s, the more accurate these averages are. We assume that S s long enough for these varables to reflect accurate averages. By usng these contnuous varables, one can provde LP relaxatons to the packet-level constrants descrbed n Secton 5. For example, by summng both sdes of Eq. () over S and nterchangng summatons between and t, one can obtan L m y, m N. Lkewse, one can obtan LP relaxatons of all the packet-level (or nstantaneous) constrants descrbed n Secton 5. For convenence, we summarze all the obtaned LP relaxaton constrants n Table (under ), Table 2 (under ), and Table 3 (under ). Table : LP relaxaton constrants under /Rado: Lm y, m N (ω β r() + )y + } j C y j ω, /NM: (ω α t() + )y + L j C + y j ω, + l C + λ l α t(), /CM: + l C µ lj β r(j), (, j) C. j y + y j λ j + µ j +, 6 LP Formulaton Let s consder a mult-hop wreless MIMO network routng nstance that conssts of a set Q of commodtes, and let x q denote lnk s data rate that belongs to commodty q. Note that the flow-

6 Table 2: LP relaxaton constrants under L m y, } m N /Rado: z α t() y, z β r() y, L /NM and /CM: y + y j, (, j) C Table 3: LP relaxaton constrants under L m y, } m N /Rado: z α t() y, z β r() y L, (Ω β r() )y + z j Ω, j C L r() /NM: L (Ω α t() )y + z j Ω, j C + L+ t() z l + θ l α t(), l L + l C + t() /CM: z l + ϑ lj β r(j), (, j) C j L + t() l L r(j) l C j z ϑ j + α t() ( y ), z j θ j + β r(j) ( y j ). balance constrants, f q f t() = s(q) x q j = j L x q j t() Otherwse, must be satsfed for all q Q and all L. By lettng { x q c = y z q Q f under f under or () (2) for all L, the mult-hop wreless MIMO network routng problem can be formulated as a standard LP whose objectve s to maxmze q Q fq subject to the flow-balance constrants gven n Eqs. () and (2), and the rado and nterference constrants gven n Table (under ), Table 2 (under ), or Table 3 (under ). 7. THROUGHPUT CHARACTERIZATION AND ANALYSIS Usng extensve smulatons, we characterze and analyze achevable throughput n mult-hop wreless MIMO networks under the three MIMO protocols (,, and ), and for the two nterference avodance models (NM and CM). Smulatons are run untl the measured throughput converges to wthn 5% of real values at a 98% confdence level. 7. The Smulaton Method and Scenaros We generate random mult-hop wreless MIMO networks, each consstng of N nodes. The medum s capacty, defned to be the maxmum number of bts that a node wth one antenna can transmt n one second, s set to unty (c =, L). All nodes are equpped wth the same number of antennas. We assume that all effectve degrees of freedom are equal to the number of antennas (α m = β m = γ m, m N). Nodes are unformly dstrbuted n a m m square where two nodes are consdered neghbors f the dstance between them does not exceed TxRange meters. For each random network, Q source-destnaton pars are randomly generated to form Q end-to-end mult-hop commodty flows. Each LP formulaton (/NM, /CM, /NM, /CM, /NM, and /CM), defned n Secton 6, s solved for each network to fnd the maxmum achevable throughput. We study the effects of the followng network parameters:. Transmsson range (TxRange): Recall that the hgher the transmsson range, the greater the nterference, but also the hgher the node degree. Typcally, a hgher nterference results n less throughput, whle a hgher node degree yelds more throughput. Here, we want to see f ths trend holds even when nodes are equpped wth MIMO lnks, and f so, to what extent t does. In ths study, we fx N to 5 and Q to 25, and vary TxRange from 6m to 32m. 2. Node densty (NodeDensty): Lke the transmsson range case, the hgher the node densty, the greater the node degree, and hence, the hgher the throughput (provded other network parameters are kept the same). Unlke the transmsson range case, ncreasng the node densty whle keepng the same number of commodtes does not, however, rase nterference levels. In ths study, we want to see how senstve throughput s to node densty when MIMO szes are vared. Here, we fx TxRange to 3 and Q to, and vary NodeDensty from % to.5% (by varyng N from 2 to 5). 3. Mult-hop length (HopLength): So far, Q source-destnaton pars are generated randomly, and hence, so are ther hop lengths (avg. hop length vared between 2.74 for TxRange = 32 and 87 for TxRange = 6). Here, we study the effect of hop length on the achevable throughput. In order to mask the effects of other network parameters, we consder a mesh network of N = 5 nodes where each node has exactly 4 neghbors. In all smulaton runs, we set the number Q of commodty flows to 25. We consder 5 dfferent hop lengths:, 3, 5, 7, and 9 hops. For each HopLength, we generate and smulate random sets, each of Q flows whose lengths are all HopLength hops. When analyzng the effects of the above parameters, we only show the results obtaned under NM; we omt those obtaned under CM as they provde smlar results (the results and analyss comparng NM wth CM are gven n Secton 7.7). The maxmum achevable throughput, shown n graphs n ths secton, are all per-commodty flow by averagng the total acheved throughput over all the Q flows. 7 Throughput Characterzaton and Analyss under Fg. 3 shows the effect of transmsson range (Fgs. 3(a) and 3(d)), node densty (Fgs. 3(b) and 3(e)), and hop length (Fgs. 3(c) and 3(f)) on the achevable throughput under. 7. The asymptotc bound Fgs. 3(a), 3(b), and 3(c) show that regardless of transmsson range, node densty, and/or hop length, as the number of antennas ncreases, the maxmum achevable throughput frst rses and then flattens out asymptotcally. Ths can be explaned as follows. Recall that multple antennas ncrease spatal reuse by allowng multple smultaneous communcaton sessons n the same vcnty,.e., nodes can, for example, use ther antennas to suppress the undesred sgnals sent by nearby transmtters, allowng them to receve nterference-free sgnals concurrently wth nearby transmtted sgnals. Therefore, one may conclude that the more antennas a

7 .5. TxRange=8 TxRange=22 TxRange=26 TxRange= (a) Effect of transmsson range NodeDensty=% NodeDensty=% NodeDensty=% NodeDensty=.5% (b) Effect of node densty HopLength=3 HopLength=5 HopLength=7 HopLength= (c) Effect of hop length.5 NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt=9 NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt=9 NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt= Transmsson Range (d) Effect of transmsson range % 5% % 5% % 5%.5% Node Densty (e) Effect of node densty Hop Length (f) Effect of hop length Fgure 3: Maxmum achevable throughput under. node has, the more nearby transmtters sgnals t can suppress, and hence, the hgher throughput the network can acheve. Because, n a gven network, each node (e.g., recever) has a fxed number of nterferng nodes (e.g., nearby transmtters), ncreasng the number of antennas beyond that fxed number of nterferng nodes cannot ncrease the throughput any further snce spatal reuse can no longer be ncreased even f more antennas are added. Ths s why we see an asymptotc bound on the achevable throughput under. 7 Effect of transmsson ranges the nterferencepath dversty tradeoff Fg. 3(a) shows that for small numbers of antennas, the hgher the transmsson range, the less the achevable throughput. Conversely, when there are a large number of antennas, the hgher the transmsson range, the greater the throughput. Also, Fg. 3(d) ndcates that as the transmsson range ncreases, the achevable throughput always decreases when each node s equpped wth a sngle antenna. In contrast, the throughput frst ncreases and then decreases when each node s equpped wth multple antennas for each MIMO sze, there exsts a transmsson range that maxmzes the achevable throughput. Note that ths optmal transmsson range ncreases as the number of antennas ncreases. Recall that n networks wth long transmsson ranges, nodes are lkely to have more neghbors. Whle ths provdes nodes wth hgher path dversty, t also provdes them wth more nterference to combat. Hence, when transmsson ranges are long, nterference domnates f nodes are only equpped wth sngle or small-szed antenna arrays whch are not enough to combat the extra nterference caused by the long ranges of transmsson, thereby achevng less overall throughput. When the number of antennas s large enough, nodes can, however, take advantage of the ncreased number of paths to fnd better routes whle effectvely combatng the nterference by usng ther antennas. In ths case, the throughput wll be ncreased as more concurrent transmssons are enabled n the same vcnty. Ths explans why for a large number of antennas, the achevable throughput for long transmsson ranges are greater than those for short transmsson ranges. 7 Effect of node densty path dversty at no nterference cost An ncrease n node densty typcally yelds path dversty as t rases the number of possble end-to-end paths. If the number Q of commodty flows s kept the same as n our case, such an ncrease n node densty does not ncur extra nterference. When the number of antennas s small ( or 2, see Fg. 3(b)), path dversty cannot be exploted to ncrease network throughput. Ths s because even when presented wth more paths to route through, nodes do not have enough antennas to suppress nterference at each of those neghborng nodes nvolved n ther mult-path routes. Ths s why the throughput achevable under small antennas szes does not depend on node densty as shown n Fg. 3(b). When the number of antennas s large, the throughput achevable n dense networks s, however, greater than that n sparse networks due to the multpath nature arsng from hgher node degrees; nodes can use ther antennas to suppress nterference at the nearby nodes nvolved n mult-path routes whle stll explotng path dversty to ncrease throughput. For each multple antenna case, Fg. 3(e) shows that there exsts a node densty beyond whch the achevable network throughput can no longer ncrease. In other words, for a gven set of commodty flows, there s a certan node densty threshold beyond whch network throughput cannot be ncreased even f nodes are provded wth more paths to route through. 7 Effect of hop length Fgs. 3(c) and 3(f) ndcate that rrespectve of the number of antennas, the larger the hop length of end-to-end flows, the less overall network throughput. Ths s because mult-hop flows wth hgh multplcty tend to create greater contenton for, and hence more nterference n, the wreless medum than those wth small hop multplcty. That s, the longer the mult-hop paths, the more flows a node s lkely to forward traffc for, and hence, the more contenton and nterference nodes are lkely to deal wth.

8 7 Throughput Characterzaton and Analyss under Fg. 4 shows the effect of transmsson range (4(a) and 4(d)), node densty (4(b) and 4(e)), and hop length (Fgs. 4(c) and 4(f)) on the maxmum achevable throughput under. These fgures ndcate that regardless of transmsson range, node densty, and/or hop length, the maxmum achevable throughput ncreases almost lnearly as a functon of the number of antennas. Unlke, under, the number of sgnals streams s proportonal to the number of antennas, and hence, so s the overall network throughput, thus makng a lnear ncrease n network throughput. Fg. 4(d) shows that the achevable throughput decreases as the transmsson range ncreases, and ths holds regardless of the sze of the antenna array. Ths declne n throughput s due to the fact that the excess of nterference resultng from the ncrease n the transmsson range cannot be suppressed under even when nodes are equpped wth many antennas; under, all antennas are exploted to ncrease data rates nstead of combatng nterference. Fg. 4(e) shows that regardless of the number of antennas, the achevable throughput also decreases as the hop length ncreases. Ths s because the ncrease n flows number of hops ntroduces extra nterference that cannot suppress, ether. Unlke the transmsson range and hop length cases, throughput does not depend on node densty, gven a fxed sze of antenna array. Ths s smply because an ncrease n node densty does not ncur extra nterference. 7 Throughput Characterzaton and Analyss under Fg. 5 shows the effect of transmsson range (5(a) and 5(d)), node densty (5(b) and 5(e)), and hop length (Fgs. 5(c) and 5(f)) on the maxmum achevable throughput under. Frst, note that the achevable throughput under ncreases almost lnearly as a functon of the number of antennas for all combnatons of transmsson range, node densty, and hop length. Recall that combnes both and n that t ncreases network throughput va spatal reuse and/or spatal multplexng, whchever provdes more overall throughput. As a result, when antennas can no longer be exploted to ncrease throughput va spatal reuse (.e., when throughput ganed va flattens out), can stll explot the antennas to ncrease network throughput further by achevng hgher data rates va spatal multplexng. 7.5 Desgn Gudelnes and Practcal Uses There s an mportant and useful trend, observed n Fg. 5(d): for each antenna array sze, there exsts an optmal transmsson range that maxmzes the achevable throughput under. For nstance, when the number of antennas s 9, the optmal transmsson range s about 22m. A smlar trend wth respect to node densty can also be observed n Fg. 5(d). Note that for every sze of antenna array, there s a certan node densty threshold beyond whch throughput can no longer be ncreased. For nstance, when the number of antennas equals 9, ths threshold s about %. Therefore, ths study can provde gudelnes for network desgners to determne optmal parameters for wreless MIMO networks; t can be used to determne optmal transmsson ranges and node denstes of wreless MIMO-equpped networks. MIMO-equpped mesh networks are an example where ths study can be very useful. For nstance, knowng the sze of antenna arrays of mesh nodes, a network desgner can use ths study to determne the optmal mesh node densty (.e., optmal number of mesh nodes) and the optmal transmsson range (.e., optmal transmsson power) that maxmze the total network throughput. 7 Spatal Reuse vs. Spatal Multplexng We now compare the performances of and aganst each other ( always outperforms the other two). Fgs. 6, 7, and 8 show throughput achevable under all MIMO protocols for dfferent values of transmsson ranges, node denstes, and hop lengths. Frst, as expected, when nodes are equpped wth sngle antennas, the achevable throughput s dentcal under all protocols, regardless of transmsson ranges, node denstes, and/or hop lengths. Second, when transmsson ranges are short (Fg. 6(a)) or node denstes are low (Fg. 7(a)), acheves hgher network throughput than that achevable under. However, when transmsson ranges or node denstes are hgh, the exact opposte trend s observed. In fact, as the transmsson range and/or the node densty ncrease, the throughput achevable under ncreases, whereas that achevable under decreases. That s, n networks wth hgh node denstes or transmsson ranges, most of the antennas are exploted to ncrease throughput va spatal reuse nstead of spatal multplexng. It can then be concluded that the antennas are frst exploted to ncrease spatal reuse by suppressng as much nterference as possble, and then the remanng antennas, f any left, are exploted to ncrease data rates va spatal multplexng. Hop lengths, on the other hand, do not affect the performances of and vs-a-vs of each other. Fg. 8 shows that the throughput achevable under s hgher than that achevable under and remans so despte the hop length. Note, however, that as the hop length ncreases, the throughput achevable under degrades more sgnfcantly than that achevable under. Ths s because greater hop lengths (.e., longer routes) typcally yeld more nterference, whch lmts the throughput obtanable under. 7.7 Non-Cooperatve vs. Cooperatve Interference Avodance Fg. 9 shows the maxmum achevable throughput under NM and CM. (Note that because NM and CM are equvalent under, we only show the results under and.) As expected, the throughput achevable under the cooperatve nterference avodance model s greater than that achevable under the non-cooperatve model. When nodes cooperate, redundant nterference suppresson can be avoded. For example, f a transmtter nterferes wth a nearby undesred recever, then both the transmtter and the recever may each end up usng one of ts antennas to avod nterference when they do not cooperate. When both the transmtter and the recever cooperate as under CM, one of them can use one of ts antennas to avod the nterference whle the other node can use ts antenna to avod nterference wth another nterferng node, thereby ncreasng the spatal reuse. Another pont worth notng s that as the number of antennas ncreases, the maxmum achevable throughput under both NM and CM converge to the same value. As explaned earler, ths s because ncreasng the number of a node s antennas above the number of ts nterferng nodes can no longer ncrease the spatal reuse regardless of the nterference model. Because network parameters such as transmsson range and node densty do not affect the obtaned comparatve results, we only showed one combnaton. 8. SUMMARY & CONCLUSIONS Ths paper models the nterference and rado constrants of multhop wreless MIMO networks under the three proposed MIMO protocols,,, and, and the two proposed nterference avodance models, NM and CM. An optmal desgn problem s

9 .5 TxRange=8. TxRange=22 TxRange=26 TxRange= (a) Effect of transmsson range NodeDensty=% NodeDensty=% NodeDensty=% NodeDensty=.5% (b) Effect of node densty HopLength=3 HopLength=5 HopLength=7 HopLength= (c) Effect of hop length NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt=9 NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt=9 NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt= Transmsson Range (d) Effect of transmsson range % 5% % 5% % 5%.5% Node Densty (e) Effect of node densty Hop Length (f) Effect of hop length Fgure 4: Maxmum achevable throughput under. TxRange=8 TxRange=22 TxRange=26 TxRange= (a) Effect of transmsson range NodeDensty=% NodeDensty=% NodeDensty=% NodeDensty=.5% (b) Effect of node densty HopLength=3 HopLength=5 HopLength=7 HopLength= (c) Effect of hop length NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt=9 2 NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt=9 NbrAnt= NbrAnt=3 NbrAnt=6 NbrAnt= Transmsson Range (d) Effect of transmsson range % 5% % 5% % 5%.5% Node Densty (e) Effect of node densty Hop Length (f) Effect of hop length Fgure 5: Maxmum achevable throughput under (a) TxRange = (b) TxRange = (c) TxRange = 28 Fgure 6: Effect of transmsson ranges on the maxmum achevable throughput under all MIMO protocols for N = 5 and Q = 25.

10 (a) NodeDensty = % (b) NodeDensty = 5% (c) NodeDensty =.5% Fgure 7: Effect of node denstes on the maxmum achevable throughput under all MIMO protocols for TxRange = 3 and Q = (a) HopLength = (b) HopLength = (c) HopLength = 9 Fgure 8: Effect of hop lengths on the maxmum achevable throughput under all MIMO protocols for N = 5 and Q = /NM /CM /NM /CM Fgure 9: Maxmum achevable throughput under and for N = 5, TxRange = 3, and Q = 25: NM vs. CM. formulated as a standard LP whose objectve s to maxmze the network throughput subject to these constrants. By solvng multple nstances of the formulated problem, we were able to characterze and analyze the maxmum achevable throughput n mult-hop wreless MIMO networks. We study the effects of several network parameters on the maxmum achevable throughput. We also llustrate how these results can be used by network desgners to determne the optmal parameters of mult-hop wreless MIMO networks. 9. REFERENCES [] R. Narasmhan, Spatal multplexng wth transmt antenna and constellaton selecton for correlated MIMO fadng channels, IEEE Tran. on Sgnal Processng, vol. 5, no., Nov. 23. [2] R. S. Blum, MIMO capacty wth nterference, IEEE Journal on Sel. Areas n Comm., vol. 2, no. 5, pp , June 23. [3] Q. H. Spencer, A. Lee Swndlehust, and M. Haardt, Zero-forcng methods for downlnk spatal multplexng n multuser MIMO channels, IEEE Tran. on Sgnal Processng, vol. 52, no. 2, Feb. 24. [4] S. K. Jayaweera and H. Vncent Poor, Capacty of multple antenna systems wth both recever and transmtter channel state nformaton, IEEE Tran. on Infor. Theory, vol. 49, no., pp , Oct. 23. [5] T. Marzetta and B. M. Hochwald, Capacty of moble multple antenna communcaton lnk n raylegh flat fadng, IEEE Tran. on Infor. Theory, vol. 45, no., pp , January 999. [6] J. C. Mundarath, P. Ramanathan, and B. D. Van Veen, A cross-layer scheme for adaptve antenna array based wreless ad hoc networks n multpath envronment, Wreless Networks (n press). [7] K. Sundaresan, R. Svakumar, M. A. Ingram, and T-Y Chang, A far medum access control protocol for ad-hoc networks wth MIMO lnks, n INFOCOM, 24. [8] A. Naspur, S. Ye, J. You, and R. E. Hromoto, A MAC protocol for moble ad hoc networks usng drectonal antennas, n WCNC, Sep. 2. [9] L. Bao and J. J. Garca-Luna-Aceves, Transmsson schedulng n ad hoc networks wth drectonal antennas, n MOBICOM, 22. [] R. R. Choudhury, X. Yang, R. Ramanathan, and N. H. Vadya, Usng drectonal antennas for medum access control n ad hoc networks, n MOBICOM, 22. [] T. Koraks, G. Jakllar, and L. Tassulas, A MAC protocol for full explotaton of drectonal antennas n ad-hoc wreless networks, n MOBIHOC, 23. [2] S. Y, Y Pe, and S. Kalyanaraman, On the capacty mprovement of ad hoc wreless networks usng drectonal antennas, n MOBIHOC, 23. [3] P. Gupta and P. R. Kumar, The capacty of wreless networks, IEEE Trans. on Infor. Theory, vol. 2, no. 46, pp , March 2. [4] A. U. Bhobe and P. L. Pern, An overvew of smart antenna technology for wreless communcatons, n IEEE Aerospace Conference., March 2. [5] M. Grossglauser and D. N. C. Tse, Moblty ncreases the capacty of ad hoc wreless networks, IEEE/ACM Transactons on Networkng, August 22. [6] P. Kyasanur and N. H. Vadya, Capacty of mult-channel wreless networks: mapct of number of channels and nterfaces, n MOBICOM, 25. [7] M. Alcherry, R. Bhata, and L. L, Jont channel assgnment and routng for throughput optmzaton n mult-channel wreless mesh networks, n MOBICOM, 25. [8] M. Kodalam and T. Nandagopal, Characterzng the capacty regon n mult-hop mult-channel wreless mesh networks, n MOBICOM, 25. [9] B. Hamdaou and P. Ramanathan, A cross-layer admsson control framework for wreless ad-hoc networks usng multple antennas, IEEE Transactons on Wreless Communcatons (n press). [2] B. Hamdaou and P. Ramanathan, Cross-layer optmzed condtons for QoS support n mult-hop wreless networks wth MIMO lnks, IEEE Journal on Selected Areas n Communcatons, May 27.